Subgroups — C₃₁×#23666 #21480425

Order 7936.

910 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2296	C₂	1	2	yes	maximal
4	C₃₁×#2088	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2087	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2086	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1735	C₂	1	2	yes	maximal
8	C₃₁×#1735	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#117	C₂²	1	2	yes	—
12	C₂×C₃₁×#117	C₂²	1	2	yes	—
13	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#427	—	2	4	no	—
15	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₃₁×#389	C₂²	1	4	yes	—
22	C₃₁×#389	C₂²	1	4	yes	—
23	C₃₁×#385	—	2	2	no	—
24	C₃₁×#385	—	2	2	no	—
25	C₄×C₄⋊C₄×C₃₁	—	2	4	no	—
26	C₃₁×#340	C₂²	1	2	yes	—
27	C₃₁×#340	C₂²	1	4	yes	—
28	C₃₁×#340	C₂²	1	4	yes	—
29	C₃₁×#340	C₂²	1	2	yes	—
30	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	—	2	4	no	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	D₄	1	4	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×#117	—	2	4	no	—
41	C₃₁×#117	—	2	4	no	—
42	C₃₁×#117	—	2	4	no	—
43	C₃₁×#117	Q₈	1	8	yes	—
44	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	—	2	2	no	—
53	C₂×C₄²×C₃₁	—	2	2	no	—
54	C₂×C₄²×C₃₁	—	2	2	no	—
55	C₂×C₄²×C₃₁	D₄	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
57	C₃₁×#107	—	2	2	no	—
58	C₃₁×#107	—	2	2	no	—
59	C₃₁×#106	—	2	4	no	—
60	C₃₁×C₄⋊C₈	—	2	4	no	—
61	C₃₁×C₄⋊C₈	—	2	4	no	—
62	C₃₁×#95	—	2	4	no	—
63	C₃₁×#95	—	2	4	no	—
64	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	4	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₃₁	#41	1	2	yes	—
78	C₂²×C₄×C₃₁	#41	1	4	yes	—
79	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₃₁	—	2	8	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	4	no	—
85	C₂×C₈×C₃₁	—	4	4	no	—
86	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	4	8	no	—
94	C₄⋊C₄×C₃₁	—	4	8	no	—
95	C₄⋊C₄×C₃₁	C₂×Q₈	1	8	yes	—
96	C₂²⋊C₄×C₃₁	—	2	8	no	—
97	C₂²⋊C₄×C₃₁	—	4	4	no	—
98	C₄²×C₃₁	—	2	4	no	—
99	C₄²×C₃₁	—	2	8	no	—
100	C₄²×C₃₁	—	2	2	no	—
101	C₄²×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	4	4	no	—
103	#23666	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	—	2	2	no	—
105	C₂³×C₃₁	—	2	2	no	—
106	C₂³×C₃₁	#121	1	2	yes	—
107	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
108	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	—	2	8	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	8	no	—
114	C₂×C₄×C₃₁	—	2	4	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	4	4	no	—
122	C₂×C₄×C₃₁	#121	1	2	yes	—
123	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
124	C₂×C₄×C₃₁	#123	1	4	yes	—
125	C₂×C₄×C₃₁	#125	1	8	yes	—
126	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
127	C₂×C₄×C₃₁	#135	1	4	yes	—
128	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
130	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
131	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
132	C₈×C₃₁	—	2	2	no	—
133	C₈×C₃₁	—	2	2	no	—
134	C₈×C₃₁	—	4	2	no	—
135	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
136	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
137	#2296	C₂×C₃₁	1	2	yes	—
138	#2088	C₂×C₃₁	1	4	yes	complemented
139	#2087	C₂×C₃₁	1	2	yes	complemented
140	#2086	C₂×C₃₁	1	2	yes	complemented
141	#1735	C₂×C₃₁	1	2	yes	—
142	#1735	C₂×C₃₁	1	1	yes	characteristic
143	C₂²×C₃₁	—	2	8	no	—
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	2	2	no	—
146	C₂²×C₃₁	#398	1	4	yes	—
147	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
148	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
149	C₂²×C₃₁	C₂×#135	1	2	yes	—
150	C₂²×C₃₁	#575	1	2	yes	—
151	C₄×C₃₁	—	2	4	no	—
152	C₄×C₃₁	—	4	8	no	—
153	C₄×C₃₁	—	4	2	no	—
154	C₄×C₃₁	—	8	2	no	—
155	C₄×C₃₁	—	8	2	no	—
156	C₄×C₃₁	#398	1	4	yes	—
157	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
158	C₄×C₃₁	C₂×#125	1	2	yes	—
159	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
161	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×#117	C₂²×C₃₁	1	2	yes	—
163	C₂×#117	C₂²×C₃₁	1	2	yes	—
164	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
165	#427	—	2	4	no	—
166	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
168	C₂×#106	C₂²×C₃₁	1	2	yes	—
169	C₂×C₄⋊C₈	—	2	4	no	—
170	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
172	#389	C₂²×C₃₁	1	4	yes	—
173	#389	C₂²×C₃₁	1	4	yes	—
174	#385	—	2	2	no	—
175	#385	—	2	2	no	—
176	C₄×C₄⋊C₄	—	2	4	no	—
177	#340	C₂²×C₃₁	1	2	yes	—
178	#340	C₂²×C₃₁	1	4	yes	—
179	#340	C₂²×C₃₁	1	4	yes	—
180	#340	C₂²×C₃₁	1	2	yes	—
181	C₂×C₃₁	—	2	4	no	—
182	C₂×C₃₁	C₂×#398	1	1	yes	characteristic
183	C₂×C₃₁	#3050	1	1	yes	characteristic
184	C₂×C₃₁	#3054	1	1	yes	characteristic
185	C₂×C₃₁	#3076	1	4	yes	—
186	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
187	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
188	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	D₄×C₃₁	1	4	yes	—
193	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
194	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
196	#117	—	2	4	no	—
197	#117	—	2	4	no	—
198	#117	—	2	4	no	—
199	#117	Q₈×C₃₁	1	8	yes	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	D₄×C₃₁	1	2	yes	—
212	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
213	#107	—	2	2	no	—
214	#107	—	2	2	no	—
215	#106	—	2	4	no	—
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	2	4	no	—
218	#95	—	2	4	no	—
219	#95	—	2	4	no	—
220	C₃₁	#23666	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
233	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
234	C₂²×C₄	#41×C₃₁	1	2	yes	—
235	C₂²×C₄	#41×C₃₁	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	8	no	—
251	C₄⋊C₄	—	4	8	no	—
252	C₄⋊C₄	C₂×Q₈×C₃₁	1	8	yes	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	8	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	4	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	C₃₁×#121	1	2	yes	—
263	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
264	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₃₁×#121	1	2	yes	—
279	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
280	C₂×C₄	C₃₁×#123	1	4	yes	—
281	C₂×C₄	C₃₁×#125	1	8	yes	—
282	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
283	C₂×C₄	C₃₁×#135	1	4	yes	—
284	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
286	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
287	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	C₃₁×#398	1	4	yes	—
295	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
296	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
297	C₂²	C₂×C₃₁×#135	1	2	yes	—
298	C₂²	C₃₁×#575	1	2	yes	—
299	C₄	—	2	4	no	—
300	C₄	—	4	8	no	—
301	C₄	—	4	2	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	C₃₁×#398	1	4	yes	—
305	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
306	C₄	C₂×C₃₁×#125	1	2	yes	—
307	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
308	C₂	—	2	4	no	—
309	C₂	C₂×C₃₁×#398	1	1	yes	characteristic
310	C₂	C₃₁×#3050	1	1	yes	characteristic
311	C₂	C₃₁×#3054	1	1	yes	characteristic
312	C₂	C₃₁×#3076	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₃₁×#2088	C₂	semidirect
5. C₃₁×#2087	C₂	semidirect
6. C₃₁×#2086	C₂	semidirect
103. #23666	C₃₁	direct (N × H)
138. #2088	C₂×C₃₁	semidirect
139. #2087	C₂×C₃₁	semidirect
140. #2086	C₂×C₃₁	semidirect
220. C₃₁	#23666	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#427	2
C₂²×C₃₁×C₄⋊C₈	2
C₃₁×#2296	2
C₃₁×#2088	2
C₃₁×#2087	2
C₃₁×#2086	2
C₃₁×#1735	2
#23666	31

Subgroups — C31×#23666 #21480425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23666 #21480425